representations of Lie algebras

First read about Lie algebras. Roughly a representation of a Lie
algebra is a way to represent that Lie algebra by linear transformations.
There is a deep and beautiful theory of such representations. I'll just scratch
the surface here by giving a precise definition and then showing a few
examples.

Let k be a field (like the real numbers or complex numbers).
If V is a k-vector space then denote by
gl(V) the collection of all linear transformations
from V to itself. This is a Lie algebra if we define
[f,g]=fg-gf, for two linear maps f,g:V-->V.
If V is finite-dimensional then gl(V) is isomorphic
as a Lie algebra to gl(n,k), which is just nxnmatrices
over k.

Definition A representation of a Lie algebra g is a Lie algebra
homomorphismH:g-->gl(V). The
dimension of the representation is the dimension of V as a vector
space. A subrepresentation is given by a subspace of V that is stable
under transformations from the image of H.

We are going to investigate this concept for everyone's favourite
Lie algebra sl(2,k) which consists of all
2x2 matrices

- -
| a b | such that a+d=0
| c d |
- -

Recall that the bracket for this Lie algebra is defined by
[A,B]=AB-BA. (The discussion doesn't depend on the field
except we do have to suppose that it has characteristic zero. Note that this is true
for C.)

To understand this Lie algebra we need to know what the brackets of these
basis elements are.[e,f] = h, [h,e] = 2e, [h,f] = -2f.
So far this is the only representation of sl(2,k) that
we know, a two-dimensional representation.

Here's another one the adjoint representation. For any Lie algebra
g we get a representation by taking V=g
and defining the representing Lie algebra homomorphism by
sending x in g to the linear transformation given
by y |--> [x,y]. Again one checks that this gives
a Lie algebra map and so this is a representation.
For sl(2,k) it is three-dimensional.

I'm am going to construct an n+1-dimensional representation
of sl(2,k), for each integer n>=0. Let V be an n+1-dimensional
vector space with basis e0,...,en.
Then define a map H:sl(2,k)-->gl(V)
as follows.

H(h)(ei)=(n-2i)ei

H(e)(ei)=(n-i+1)ei-1

H(f)(ei)=(i+1)ei+1

for 0<=i<=n. (We define e-1=en+1=0.)
Checking that H is a Lie algebra homomorphism is easy. For example,
[H(e),H(f)](ei)=H(e)H(f)(ei)-H(f)H(e)(ei)
by definition. Applying the formulae above we get that
the right hand side is H(e)((i+1)ei+1)-H(f)((n-i+1)ei-1).
Apply the formulae once more and this is
(n-i)(i+1)ei-(n-i+1)iei=(n-2i)ei.
This is H([e,f]), as required. Checking the other
brackets is similar.

It turns out that these representations are irreducible (they have no
subrepresentations except the zero space and the whole representation)
and that every representation of sl(2,k) may be contructed
by direct sums of these representations.

This kind of thing is decpetively useful for dealing with angular momentum in Quantum Mechanics. If you look at the main write up there, the algebra works out exactly the same if you let:

h = 2J3/(h-bar)

e = J+/(h-bar)

f = J-/(h-bar)

...and then all the commutation relations come out analagously. In fact, there is another operator useful in representing sl(2,C), called the Cazimir operator; Ϊ = ef + fe + (1/2)h2. Then given this, it's not hard to show that Ϊ = J2/(h-bar)2.